\(\int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx\) [102]

Optimal result

Integrand size = 27, antiderivative size = 138 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {b x^2 \sqrt {d-c^2 d x^2}}{4 c d \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 c^2 d}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b c^3 d \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

1/4*b*x^2*(-c^2*d*x^2+d)^(1/2)/c/d/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*x*(-c^2 
*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c^2/d-1/4*(-c^2*d*x^2+d)^(1/2)*(a+b*arc 
cosh(c*x))^2/b/c^3/d/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.02 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-\frac {4 a c x \sqrt {d-c^2 d x^2}}{d}-\frac {4 a \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{\sqrt {d}}+\frac {b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (-\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)+\sinh (2 \text {arccosh}(c x))))}{\sqrt {d-c^2 d x^2}}}{8 c^3} \] Input:

Integrate[(x^2*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]
 

Output:

((-4*a*c*x*Sqrt[d - c^2*d*x^2])/d - (4*a*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/ 
(Sqrt[d]*(-1 + c^2*x^2))])/Sqrt[d] + (b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c* 
x)*(-Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] + Sinh[2*ArcCosh[ 
c*x]])))/Sqrt[d - c^2*d*x^2])/(8*c^3)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6353, 15, 6307}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx\)

\(\Big \downarrow \) 6353

\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \int xdx}{2 c \sqrt {d-c^2 d x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 c^2 d}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 c^2 d}-\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1}}{4 c \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 6307

\(\displaystyle -\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 c^2 d}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{4 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1}}{4 c \sqrt {d-c^2 d x^2}}\)

Input:

Int[(x^2*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]
 

Output:

-1/4*(b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*Sqrt[d - c^2*d*x^2]) - (x*Sqr 
t[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*c^2*d) + (Sqrt[-1 + c*x]*Sqrt[1 
+ c*x]*(a + b*ArcCosh[c*x])^2)/(4*b*c^3*Sqrt[d - c^2*d*x^2])
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 6307

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d 
 + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x 
] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
 

rule 6353

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcCosh[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2 
*p + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] 
- Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x) 
^p)]   Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*Ar 
cCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2* 
d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(299\) vs. \(2(118)=236\).

Time = 0.27 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.17

method	result	size
default	\(-\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{4 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(300\)
parts	\(-\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{4 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(300\)

Input:

int(x^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*a*x/c^2/d*(-c^2*d*x^2+d)^(1/2)+1/2*a/c^2/(c^2*d)^(1/2)*arctan((c^2*d) 
^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(-1/4*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2 
)*(c*x+1)^(1/2)/d/c^3/(c^2*x^2-1)*arccosh(c*x)^2-1/16*(-d*(c^2*x^2-1))^(1/ 
2)*(2*c^3*x^3-2*c*x+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c 
*x+1)^(1/2))*(-1+2*arccosh(c*x))/d/c^3/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^( 
1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+ 
1)^(1/2)-2*c*x)*(1+2*arccosh(c*x))/d/c^3/(c^2*x^2-1))
 

Fricas [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:

integrate(x^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas 
")
 

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(b*x^2*arccosh(c*x) + a*x^2)/(c^2*d*x^2 - d 
), x)
 

Sympy [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \] Input:

integrate(x**2*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**(1/2),x)
 

Output:

Integral(x**2*(a + b*acosh(c*x))/sqrt(-d*(c*x - 1)*(c*x + 1)), x)
 

Maxima [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:

integrate(x^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima 
")
 

Output:

-1/2*a*(sqrt(-c^2*d*x^2 + d)*x/(c^2*d) - arcsin(c*x)/(c^3*sqrt(d))) + b*in 
tegrate(x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c^2*d*x^2 + d), x 
)
 

Giac [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:

integrate(x^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")
 

Output:

integrate((b*arccosh(c*x) + a)*x^2/sqrt(-c^2*d*x^2 + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:

int((x^2*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(1/2),x)
 

Output:

int((x^2*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\mathit {asin} \left (c x \right ) a -\sqrt {-c^{2} x^{2}+1}\, a c x +2 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{2}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{3}}{2 \sqrt {d}\, c^{3}} \] Input:

int(x^2*(a+b*acosh(c*x))/(-c^2*d*x^2+d)^(1/2),x)
 

Output:

(asin(c*x)*a - sqrt( - c**2*x**2 + 1)*a*c*x + 2*int((acosh(c*x)*x**2)/sqrt 
( - c**2*x**2 + 1),x)*b*c**3)/(2*sqrt(d)*c**3)